{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nThe [[Möbius function|0C8QENHYZW]] $\\mu$ is a [[multiplicative function|0AP80TT660]].\nThat is, $\\mu(1) = 1$ and $\\mu(mn) = \\mu(m) \\mu(n)$ for all coprime positive integers $m$ and $n$.\n\n## Remarks\n\nThe function $\\mu$ is not completely multiplicative: $\\mu(4) = 0$, but $\\mu(2)^2 = 1$.\nThe proof: for coprime $m$ and $n$, the product $mn$ is squarefree exactly when $m$ and $n$ are both squarefree. Then $\\omega(mn) = \\omega(m) + \\omega(n)$.\n\n## Explanations\n\n### L1 human\n\nFor example, $\\mu(5) = -1$ and $\\mu(6) = 1$, and $\\mu(30) = -1 = \\mu(5) \\mu(6)$.\nWhen $m$ and $n$ have no common prime, their prime factors just combine, so the signs multiply.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "11A25"
    ],
    "name": "Multiplicativity of the Möbius function",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/M%C3%B6bius_function"
      }
    ],
    "tag": "0VB9G5B034",
    "uses": [
      "0C8QENHYZW",
      "0AP80TT660"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0VB9G5B034/",
  "links": [
    {
      "field": "uses",
      "target": "0C8QENHYZW"
    },
    {
      "field": "uses",
      "target": "0AP80TT660"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0VB9G5B034.md",
  "msc": [
    "11A25"
  ],
  "name": "Multiplicativity of the Möbius function",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0VB/0VB9G5B034.md",
  "status": null,
  "tag": "0VB9G5B034",
  "uri": "m2://0VB9G5B034",
  "url": "https://math2.tamnd.com/wiki/multiplicativity-of-the-mobius-function/",
  "writing": {
    "tag": "0VB9G5B034",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}